Deck 7: Analytic Geometry and Nonlinear Systems

Question 1

Consider the system of equations
$3 x^{2}+y^{2}=12$
$2 x+2 y=-4$
(a) What type of graph does each equation have?
(b) How many points of intersection are possible for these types of graphs?
(c) Solve the system.
(d) Support the solutions using a graphing calculator.

Answer 1

(a) The first is the equation of an ellipse; the second is the equation of a line.
(b) 0,1 , or 2
(c) $(1,-3),(-2,0)$
(d)

Question 2

Solve the following system using one of the row echelon methods.
3 x+y-z =-2
-2 x+6 y+5 z =1
x+2 y+2 z =3

Answer 2

$(1,-2,3)$

Question 3

Perform the following matrix operations if possible.
(a) $\left[\begin{array}{rr}1 & -2 \\ 4 & 7 \\ -3 & 0\end{array}\right]-\left[\begin{array}{rr}0 & 7 \\ 5 & 3 \\ -5 & 8\end{array}\right]$
(b) $\left[\begin{array}{rrr}6 & 2 & -3 \\ 7 & -9 & 10\end{array}\right]+\left[\begin{array}{rr}5 & -4 \\ 3 & 8\end{array}\right]$
(c) $\left[\begin{array}{rr}2 & -1 \\ 0 & 3 \\ 1 & 5\end{array}\right]\left[\begin{array}{rrr}5 & -2 & 3 \\ -3 & 4 & -1\end{array}\right]$

Answer 3

(a) $\left[\begin{array}{rr}1 & -9 \\ -1 & 4 \\ 2 & -8\end{array}\right]$
(b) Not possible
(c) $\left[\begin{array}{rrr}13 & -8 & 7 \\ -9 & 12 & -3 \\ -10 & 18 & -2\end{array}\right]$

Question 4

Suppose that $A$ is a $m \times 3$ matrix and $B$ is a $4 \times m$ matrix.
(a) Can $A B$ be found? If so, give the dimensions of the resulting matrix.
(b) Can $B A$ be found? If so, give the dimensions of the resulting matrix.
(c) Does $A B=B A$ ? Explain why or why not.
(d) If $C$ and $D$ are each square matrices, will $C D$ always exist?

Question 5

Evaluate each determinant.
(a) $\operatorname{det}\left[\begin{array}{rr}3 & 11 \\ -2 & 8\end{array}\right]$
(b) $\operatorname{det}\left[\begin{array}{rrr}3 & 2 & -2 \\ 1 & -3 & 4 \\ 2 & 5 & 1\end{array}\right]$

Question 6

Solve the system by using Cramer's rule.
3 x+2 y =8
-5 x+4 y =-17

Question 7

Consider the system of equations
$$\begin{aligned}x+3 y+z & =3 \\2 x+7 y & =16 \\-3 x-7 y-6 z & =7\end{aligned}$$
(a) Write the matrix of coefficients $A$ , the matrix of variables $X$ , and the matrix of constants $B$ for this system.
(b) Find $A^{-1}$ .
(c) Use the matrix inverse method to solve the system.
(d) If the matrix of constants $B$ is replaced by the matrix $\left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]$ , find the solution to $A X=B$ .

Question 8

A small city conducts a population census every two years; the census was begun in the year 2000. The population declined in the early 2000s but began to grow in later years. The table shows the population, in hundred thousands, for several years with 2000 represented by year 0,2002 by year 2 , and so on. Use the data points $(0,2.21),(4,1.95)$ , and $(8,2.30)$ to find a quadratic function defined by $f(x)=a x^{2}+b x+c$ that models the data. Graph $f$ together with the data.

Question 9

The solution set of a system of inequalities is shown below. Which of the following systems is it?

(a) $y+x \leq 7$
$y \geq x^{2}-4 x+3$
(b) $y+x \geq 7$
$y \geq x^{2}-4 x+3$
(c) $\begin{aligned} & y+x \geq 7 \\ & y \leq x^{2}-4 x+3\end{aligned}$
(d) $y+x \leq 7$
$y \leq x^{2}-4 x+3$

Question 10

A restaurant supply company must ship at least 40 cases of vegetables to two midwest restaurants. Each restaurant can accommodate at most 40 cases of vegetables. Restaurant A already has 10 cases of vegetables on hand, while restaurant $\mathrm{B}$ already has 8 cases. It costs $\$ 4.80$ to ship a case of vegetables to restaurant $\mathrm{A}$ and $\$ 4$ to ship a case of vegetables to restaurant $B$ . How many cases should be shipped to each restaurant to minimize cost? What is the minimum cost?

Question 11

Find the partial fraction decomposition for the rational expression $\frac{x-11}{x^{2}-2 x-3}$ .

Question 12

Find the partial fraction decomposition for the rational expression $\frac{5 x^{2}+2 x+1}{(x-1)(x+1)^{2}}$ .

Question 13

Consider the system of equations
$4 x^{2}+y^{2}=20$
$-2 x+3 y=10$
(a) What type of graph does each equation have?
(b) How many points of intersection are possible for these types of graphs?
(c) Solve the system.
(d) Support the solutions using a graphing calculator.

Question 14

Solve the following system using one of the row echelon methods.
$\begin{aligned}x+y+2 z & =4 \\3 x-2 y+4 z & =12 \\2 x-2 y+z & =1\end{aligned}$

Question 15

Perform the following matrix operations if possible.
(a) $\left[\begin{array}{rr}5 & -3 \\ 7 & -2 \\ 0 & 4\end{array}\right]\left[\begin{array}{rr}1 & 2 \\ -1 & 6 \\ 3 & 3\end{array}\right]$
(b) $\left[\begin{array}{rr}1 & -3 \\ 2 & 0 \\ 4 & -5\end{array}\right]-3\left[\begin{array}{rr}-1 & 4 \\ 3 & 4 \\ 2 & -2\end{array}\right]$
(c) $\left[\begin{array}{rr}2 & 0 \\ -3 & -1 \\ 4 & 6\end{array}\right]\left[\begin{array}{rrr}1 & 2 & 3 \\ -1 & 0 & -2\end{array}\right]$

Question 16

Suppose that $A$ is a $3 \times m$ matrix and $B$ is a $m \times 2$ matrix.
(a) Can $A B$ be found? If so, give the dimensions of the resulting matrix.
(b) Can $B A$ be found? If so, give the dimensions of the resulting matrix.
(c) Does $A B=B A$ ? Explain why or why not.
(d) If $C$ and $D$ are each square matrices, will $C D$ always exist?

Question 17

Evaluate each determinant.
(a) $\operatorname{det}\left[\begin{array}{rr}-6 & 9 \\ 4 & -5\end{array}\right]$
(b) $\operatorname{det}\left[\begin{array}{rrr}2 & 0 & -2 \\ 3 & 1 & 5 \\ 6 & 4 & 3\end{array}\right]$

Question 18

Solve the system by using Cramer's rule.
$\begin{aligned}& 7 x+2 y=-4 \\& 4 x+3 y=7\end{aligned}$

Question 19

Consider the system of equations
$$\begin{aligned}x-3 y+2 z & =5 \\2 x-4 y+3 z & =12 \\3 x-7 y+6 z & =23\end{aligned}$$
(a) Write the matrix of coefficients $A$ , the matrix of variables $X$ , and the matrix of constants $B$ for this system.
(b) Find $A^{-1}$ .
(c) Use the matrix inverse method to solve the system.
(d) If the matrix of constants $B$ is replaced by the matrix $\left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]$ , find the solution to $A X=B$ .

Question 20

The minimum yearly temperature in a certain temperate climate was recorded each year for five years beginning
in 2000. The minimum temperature increased for the first few years before declining. The table shows the temperature, in degrees Fahrenheit, for those years, with 2000 represented by year 0, 2001 by year 1, and so on. Use the data points $(0,38.8),(2,47.3)$ , and $(4,39)$ to find a quadratic function defined by $f(x)=a x^{2}+b x+c$ that models the data. Graph $f$ together with the data.

Question 21

The solution set of a system of inequalities is shown below. Which of the following systems is it?


(a) $y-2 x \leq-9$
$y \leq-x^{2}+6 x-4$
(b) $y-2 x \leq-9$
$y \geq-x^{2}+6 x-4$
(c) $y-2 x \geq-9$
$y \leq-x^{2}+6 x-4$
(d) $y-2 x \geq-9$
$y \geq-x^{2}+6 x-4$

Question 22

A restaurant supply company must ship at least 80 cases of seafood to two midwest restaurants. Each restaurant can accommodate at most 80 cases of seafood. Restaurant A already has 20 cases of seafood on hand, while restaurant B already has 16 cases. It costs $\$ 9.60$ to ship a case of seafood to restaurant A and $\$ 8$ to ship a case of seafood to restaurant B. How many cases should be shipped to each restaurant to minimize cost? What is the minimum cost?

Question 23

Find the partial fraction decomposition for the rational expression $\frac{5 x+11}{x^{2}+5 x+6}$ .

Question 24

Find the partial fraction decomposition for the rational expression $\frac{-2 x^{2}-2 x+48}{(x+3)(x-3)^{2}}$ .

Question 25

Consider the system of equations
$3 x^{2}-4 y=12$
$-9 x+2 y=-18$
(a) What type of graph does each equation have?
(b) How many points of intersection are possible for these types of graphs?
(c) Solve the system.
(d) Support the solutions using a graphing calculator.

Question 26

Solve the following system using one of the row echelon methods.
$\begin{array}{r}x-3 y+2 z=1 \\2 x-4 y+3 z=3 \\3 x-7 y+6 z=3\end{array}$

Question 27

Perform the following matrix operations if possible.
(a) $\left(\begin{array}{lll}1 & -7 & 3\end{array}\right)-\left(\begin{array}{ll}4 & -5\end{array}\right)$
(b) $\left[\begin{array}{rrr}1 & 6 & 7 \\ 0 & -3 & 5\end{array}\right]-2\left[\begin{array}{rrr}6 & -5 & 2 \\ -1 & 0 & 3\end{array}\right]$
(c) $\left(\begin{array}{lll}4 & 3 & -2\end{array}\right)\left[\begin{array}{rr}1 & -2 \\ -1 & 5 \\ 6 & 0\end{array}\right]$

Question 28

Suppose that $A$ is a $n \times 2$ matrix and $B$ is a $2 \times n$ matrix.
(a) Can $A B$ be found? If so, give the dimensions of the resulting matrix.
(b) Can $B A$ be found? If so, give the dimensions of the resulting matrix.
(c) Does $A B=B A$ ? Explain why or why not.
(d) If $C$ and $D$ are each square matrices, will $C D$ always exist?

Question 29

Evaluate each determinant.
(a) $\operatorname{det}\left[\begin{array}{rr}6 & 3 \\ -8 & 2\end{array}\right]$
(b) $\operatorname{det}\left[\begin{array}{rrr}3 & 1 & -2 \\ 1 & -4 & 3 \\ 7 & 11 & -12\end{array}\right]$

Question 30

Solve the system by using Cramer's rule.
$5 x+9 y=2$
$7 x-18 y=-4$

Question 31

Consider the system of equations
$x+2 y-3 z=8$
$2 x+5 y-11 z=18$
$3 x+7 y-13 z=24$
(a) Write the matrix of coefficients $A$ , the matrix of variables $X$ , and the matrix of constants $B$ for this system.
(b) Find $A^{-1}$ .
(c) Use the matrix inverse method to solve the system.
(d) If the matrix of constants $B$ is replaced by the matrix $\left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]$ , find the solution to $A X=B$ .

Question 32

The employment rate in a city has been documented at two year intervals. The table shows the number of employed citizens, in millions, tor several years, with 2000 represented by year 0,2002 by year 2 , and so on. Use the data points $(0,2.9),(4,2.3)$ , and $(8,3.0)$ to find a quadratic function defined by $f(x)=a x^{2}+b x+c$ that models the data. Graph $f$ together with the data.

Question 33

The solution set of a system of inequalities is shown below. Which of the following systems is it?

(a) $2 y-3 x \geq 8$
$y \leq x^{2}+6 x+7$
(b) $2 y-3 x \leq 8$
$y \leq x^{2}+6 x+7$
(c) $2 y-3 x \geq 8$
$y \geq x^{2}+6 x+7$
(d) $2 y-3 x \leq 8$
$y \geq x^{2}+6 x+7$

Question 34

A restaurant supply company must ship at least 120 cases of cheese to two midwest restaurants. Each restaurant can accommodate at most 120 cases of cheese. Restaurant A already has 30 cases of cheese on hand, while restaurant B already has 24 cases. It costs $\$ 14.40$ to ship a case of cheese to restaurant A and $\$ 12$ to ship a case of cheese to restaurant B. How many cases should be shipped to each restaurant to minimize cost? What is the minimum cost?

Question 35

Find the partial fraction decomposition for the rational expression $\frac{4 x+7}{x^{2}-9 x+14}$ .

Question 36

Find the partial fraction decomposition for the rational expression $\frac{5 x^{2}+30 x+56}{(x-4)(x+4)^{2}}$ .

Question 37

Consider the system of equations
$$\begin{aligned}3 x^{2}+4 y & =12 \\-9 x+2 y & =-18\end{aligned}$$
(a) What type of graph does each equation have?
(b) How many points of intersection are possible for these types of graphs?
(c) Solve the system.
(d) Support the solutions using a graphing calculator.

Question 38

Solve the following system using one of the row echelon methods.
$\begin{aligned}x+5 y+z & =-7 \\4 x+2 y+7 z & =32 \\2 x-3 y-z & =19\end{aligned}$

Question 39

Perform the following matrix operations if possible.
(a) $\left[\begin{array}{rr}3 & -1 \\ 5 & 2\end{array}\right]\left[\begin{array}{rrr}1 & -4 & 1 \\ 3 & 2 & -2\end{array}\right]$
(b) $\left[\begin{array}{rrr}2 & -2 & 6 \\ 1 & 0 & -7\end{array}\right]+3\left[\begin{array}{rrr}-1 & 1 & -2 \\ 1 & 3 & 4\end{array}\right]$
(c) $\left[\begin{array}{rrr}3 & -4 & 5 \\ 7 & 10 & -4\end{array}\right]\left[\begin{array}{rrr}5 & 2 & -9 \\ 0 & 1 & 5\end{array}\right]$

Question 40

Suppose that $A$ is a $1 \times n$ matrix and $B$ is a $n \times 1$ matrix.
(a) Can $A B$ be found? If so, give the dimensions of the resulting matrix.
(b) Can $B A$ be found? If so, give the dimensions of the resulting matrix.
(c) Does $A B=B A$ ? Explain why or why not.
(d) If $C$ and $D$ are each square matrices, will $C D$ always exist?

Question 41

Evaluate each determinant.
(a) $\operatorname{det}\left[\begin{array}{rr}8 & 6 \\ -2 & -7\end{array}\right]$
(b) $\operatorname{det}\left[\begin{array}{rrr}1 & 3 & 4 \\ -2 & 4 & 5 \\ 7 & 6 & -3\end{array}\right]$

Question 42

Solve the system by using Cramer's rule.
$3 x+12 y=5$
$9 x-8 y=4$

Question 43

Consider the system of equations
$$\begin{array}{r}x+4 y-2 z=0 \\-2 x-6 y+5 z=5 \\3 x+14 y-4 z=8\end{array}$$
(a) Write the matrix of coefficients $A$ , the matrix of variables $X$ , and the matrix of constants $B$ for this system.
(b) Find $A^{-1}$ .
(c) Use the matrix inverse method to solve the system.
(d) If the matrix of constants $B$ is replaced by the matrix $\left[\begin{array}{l}0 \\ 0 \\ 0\end{array}\right]$ , find the solution to $A X=B$ .

Question 44

During one period, the average price of a gallon of gasoline in a town changed from day to day. The table shows the average price, in dollars, for each of several consecutive days, where 0 represents Sunday, 1 represents Monday, and so on. Use the data points $(0,3.50),(2,3.15)$ , and $(4,3.6)$ to find a quadratic function defined by $f(x)=a x^{2}+b x+c$ that models the data. Graph $f$ together with the data.

Question 45

The solution set of a system of inequalities is shown below. Which of the following systems is it?

(a) $2 y-x \leq-4$
$y \geq-x^{2}-8 x-9$
(b) $2 y-x \geq-4$
$y \geq-x^{2}-8 x-9$
(c) $2 y-x \leq-4$
$y \leq-x^{2}-8 x-9$
(d) $2 y-x \geq-4$
$y \leq-x^{2}-8 x-9$

Question 46

A restaurant supply company must ship at least 60 cases of lettuce to two midwest restaurants. Each restaurant can accommodate at most 60 cases of lettuce. Restaurant A already has 15 cases of lettuce on hand, while restaurant B already has 12 cases. It costs $\$ 7.20$ to ship a case of lettuce to restaurant A and $\$ 6$ to ship a case of lettuce to restaurant B. How many cases should be shipped to each restaurant to minimize cost? What is the minimum cost?

Question 47

Find the partial fraction decomposition for the rational expression $\frac{7 x+1}{x^{2}-x-20}$ .

Question 48

Find the partial fraction decomposition for the rational expression $\frac{x^{2}+9 x-18}{(x+2)(x-2)^{2}}$ .